Aim of the article and pre-requisites

By the end of this hands-on tutorial you should be familiar with:

The minimum knowledge to benefit from the tutorial includes being familiar with:

Familiarity with systematic reviews

You are leading a research project involving a systematic review of multiple interventions for a specific health condition and patient population. You have put in a lot of effort to establish a high-standards protocol, registered it with PROSPERO register and completed the data extraction.

While reviewing the extracted data, you notice that many eligible trials report having missing participants. How are you going to proceed with the analysis?

The good, the bad, and the ugly

Consider the small fictional trial in Figure 1, where 20 participants have been randomised into two interventions and the outcome is binary (symptoms improved or worsened after intervetion). Some participants completed the trial (completers), while others left the trial prematurely for several reasons (missing). The goal is to draw conclusions using the entire randomised sample.

Figure 1: A fictional small trial of two interventions

Without individual participant data, there are only a few methods to handle missing participants in (network) meta-analysis:

  1. Exclusion – under the assumption that the missing participants’ outcomes are missing due to reasons unrelated to the planning and conduct of the trial (missing completely at random, MCAR), or due to their baseline characteristics (missing at random, MAR),
  2. Imputation – assuming that the missing participants would have improved or worsened symptoms had they not left the trial (missing not at random, MNAR),
  3. Modelling – the missingness mechanism (i.e., M(C)AR or MNAR) using a probability distribution to describe the association between the missing and completers’ outcomes (pattern-mixture model).

Exclusion

Both exclusion and imputation are suboptimal methods in handling missing participants. Exclusion, as shown in Figure 2, is not appropriate as it reduced the sample to those participants who remained in the trial and decreases the power to detect significant associations. Moreover, if the missingness mechanism is not M(C)AR, the risk of estimating a biased intervention effect (e.g., odds ratio) is imminent.

Figure 2: Excluding missing participants from both interventions

Imputation

Figure 3 illustrates the imputation of missing participants in two interventions. In in the first intervention, it is assumed that all missing participants experienced symptom improvement and thus, left the trial prematurely. In the second intervention, it is assumed that all missing participants experienced symptom deterioration and left the trial prematurely. Imputation implies adding the number of missing participants to the outcome based on the corresponding scenario: to the number of events (improvement) in the first intervention and to the number of non-events (deterioration) in the second intervention. This method preserves the randomised sample but assumes the missingness scenario. Without following the missing participants to record their outcome, we cannot know the true missingness mechanism. Hence, any assumptions we make about the missingness mechanism should naturally propagate in the estimation of the intervention effect by increasing its standard error. Imputation rears its ugly head by ‘treating’ the missing participants as observed, increasing the risk to estimate a biased intervention effect and draw erroneous statistical significance conclusions.

Figure 3: Imputing missing participants making different scenarios in each intervention

(Network) meta-analysis inherits the limitations of exclusion and imputation observed at trial-level via the inclusion of trials with missing participants. As you do not have individual participant data to make use of different cool methods for missing participants, the quality of the (network) meta-analysis results will strongly depend, among others things, on your good guess about the missingness mechanism and how you incorporate this good guess into the results. We have good news on that; just, keep reading.

Pattern-mixture model

The last member of the squad of methods for missing participants in (network) meta-analysis is the pattern-mixture model. The pattern-mixture model elegantly adjusts, rather than fixes, the average outcome (in this case, the risk of event) to the assumed missingness mechanism, and propagates this assumption to the standard error of the estimated intervention effect, while maintaining the randomised sample. It allows for different assumptions about the missing mechanism between interventions or across the trials, offering a flexible framework for properly handling missing participants in (network) meta-analysis. If exclusion, imputation and pattern-mixture model were characters from the epic Western spaghetti film, The Good, the Bad and the Ugly, the correspondence of who would be who is rather clear.

Figure 4 shows two normal distributions of the informative missingness odds ratio (IMOR), which is a central parameter in the pattern-mixture model for a binary outcome. IMOR is similar to the odds ratio for two interventions, as it compares the odds of an event in missing participants with the odds of an event in completers. An IMOR equal to 1 indicates the MAR mechanism (i.e., no difference in the compared groups), and an IMOR different from 1 implies the MNAR mechanism (i.e., the compared groups differ). To illustrate this, we have considered a different IMOR distribution for each intervention. For the the first intervention, we assumed that, on average, the odds of an event
in completers were twice that in missing participants (the mode), with other scenarios being less likely to occur as we deviate from the mode. For the second intervention, we assumed that, on average, the odds of an event in missing participants were twice that in completers (the mode), again, with other scenarios being less likely to occur as we deviate from the mode. It is important to note that the normal distributions presented in Figure 4 are implied for the IMOR in the logarithmic scale, which is commonly used when estimating the odds ratio.

Figure 4: The IMOR parameter under different scenarios for each intervention

The versatility of the rnmamod R package

The rnmamod R package was originally developed to address the lack of dedicated functions for handling missing participants in pairwise and network meta-analysis in existing R packages. Over time, our goal expanded to create a comprehensive suite of functions for performing and visualizing Bayesian pairwise and network meta-analysis. The package implements the pattern-mixture model for Proper handling of missing participants in all models. In this tutorial, we will walk you through the core functions of the package and provide a comprehensive visual summary of the results.

Let’s get started

You may install and load the package directly from CRAN by running the code below:

install.packages("rnmamod")
library(rnmamod)

However, we recommend installing the development version to experience the latest advances in the package:

install.packages("devtools")
devtools::install_github("LoukiaSpin/rnmamod")

We will use the dataset of Baker et al. (2009), which includes 21 trials comparing seven pharmacological interventions and placebo in chronic obstructive pulmonary disease (COPD) patients. The aim is to determine if patients experienced COPD exacerbation after receiving the randomised intervention. Run head(nma.baker2009) to see the first six trials, or nma.baker2009 to see the entire dataset. The dataset has one-trial-per-row format, which is the typical format encountered in BUGS language models.

head(nma.baker2009)
#>                 study t1 t2 t3 t4 r1 r2 r3 r4 m1 m2 m3 m4  n1  n2 n3 n4
#> Llewellyn-Jones, 1996  1  4 NA NA  3  0 NA NA  1  0 NA NA   8   8 NA NA
#>        Paggiaro, 1998  1  4 NA NA 51 45 NA NA 27 19 NA NA 139 142 NA NA
#>          Mahler, 1999  1  7 NA NA 47 28 NA NA 23  9 NA NA 143 135 NA NA
#>        Casaburi, 2000  1  8 NA NA 41 45 NA NA 18 12 NA NA 191 279 NA NA
#>       van Noord, 2000  1  7 NA NA 18 11 NA NA  8  7 NA NA  50  47 NA NA
#>         Rennard, 2001  1  7 NA NA 41 38 NA NA 29 22 NA NA 135 132 NA NA

Network plot and evidence description

To create a network plot use the netplot function (type ?netplot to learn more):

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# Create the network plot and tables
netplot(data = nma.baker2009, 
        drug_names = interv_names, 
        text.cex = 1.5)

The function currently calls the nma.networkplot function from the pcnetmeta R package. The netplot function gives further insights into the network evidence by printing the following three tables on the console:

  • A description of the network evidence

  • A summary of the trials, randomised sample, and outcome data for each intervention

* A summary of the trials, randomised sample, and outcome data for each observed comparison in the network

Mapping the evidence on missing participants

The nma.baker2009 dataset was selected due to its high number of missing participants in most trials, interventions and comparisons. Use the heatmap_missing_dataset function to view the distribution of missing participants (in percent) in the dataset. Most trials have interventions with high level of missingness, with only a few trials having zeror or low missingness rate.

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# Create the heatmap of % missing participants for each trial and arm
heatmap_missing_dataset(data = nma.baker2009, 
                        trial_names = nma.baker2009$study,
                        drug_names = abbr_interv_names)

Use the heatmap_missing_network function to view the median and range of missing participants (in percent) for each intervention (on the main diagonal) and comparison (lower off-diagonal):

# Abbreviated interventions (in the order they appear in the dataset)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# Create the heatmap of % missing participants for each intervention and observed comparison
heatmap_missing_network(data = nma.baker2009, 
                        drug_names = abbr_interv_names)

We can immediately spot the comparison with the lowest median of missing participants: tiotropium versus formoterol.

The fine architecture of the rnmamod package

This section introduces you to the backbone function of the rnmamod architecture: the run_model. This function conducts Bayesian network meta-analysis while accounting for missing participants using the pattern-mixture model. Most functions in the package cannot work without the run_model function and, on its own, it returns results for a bulk of model parameters. The magic happens once the run_model function is fed into the other functions of the package, which enable you to run all necessary analyses and obtain various graphs to understand and interpret the results.

Conducting a random-effects network meta-analysis

Let’s run a Bayesian random-effects (model = "RE") network meta-analysis while addressing missing participants reported in each intervention arm of every trial. The effect measure of interest is the odds ratio (measure = "OR"). A half-normal prior distribution is used for the between-trial standard deviation with scale parameter equal to 1 (heter_prior = list("halfnormal", 0, 1)) and one IMOR parameter is estimated for each intervention in the network (assumption = "IDE-ARM"). The model operates under the MAR assumption on average (mean_misspar = c(0, 0)) with a variance equal to 1 (var_misspar = 1) for the outcome being harmful (D = 0). Assign a name to the function to use it as an object in the other functions in the model. The run_model function and all other model functions run in JAGS.

## Run a random-effects network meta-analysis
res <- run_model(data = nma.baker2009,
                 measure = "OR",
                 model = "RE",
                 assumption = "IDE-ARM",
                 heter_prior = list("halfnormal", 0, 1),
                 mean_misspar = c(0, 0), 
                 var_misspar = 1,
                 D = 0,
                 n_chains = 3,
                 n_iter = 10000,
                 n_burnin = 1000,
                 n_thin = 1)

On the console you see will two sets of progress bars: the first one shows the progress of the simulation. Once the model completes the iterations for all chains, a second progress bar appears on the console that shows the progress of the model update for n_iter = 10000 iterations until convergence:

Specifying carefully the arguments of the run_model function is essential for the subsequent analyses to yield sensible results. This requires thorough consideration of the proper effect measure (measure), the meta-analysis model (model), the structure of the missingness parameter (assumption), the prior distribution for the heterogeneity parameter (heter_prior), the missingness mechanism (mean_misspar and var_misspar), the direction of the outcome (D), and the necessary tuning parameters to run Markov chain Monte Carlo simulations using JAGS (n_chains, n_iter, n_burning, and n_thin). Refer to the documentation of the function by typing ?run_model.

Now we can use res in the arguments of the other model functions to conduct the subsequent analyses, and visualize the results. The journey has just begun.

Markov chain Monte Carlo diagnostics

Use the mcmc_diagnostic function, to check the model’s convergence.

# Display diagnostics for Bayesian methods
mcmc_diagnostics(net = res,
                 par = c("EM", "tau"))

Here, we illustrate a panel of bar plots on the log odds ratio for all possible comparisons in the network (x-axis):

It displays diagnostic plots (trace, density, and autocorrelation) for the monitored parameters specified in the par argument (here, the log odds ratio for all possible pairwise comparisons, "EM", and the common between-trial standard deviation, "tau"). Access all monitored parameters from run_model using res$jagsfit.

The league tables

Estimation

Use the league_heatmap function to create the popular league table:

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# The heatmap league table for estimation 
league_heatmap(full1 = res, 
               drug_names1 = interv_names)

The table is read as row versus column and shows the (posterior mean) odds ration and 95% credible interval for each comparison The interventions are sorted in decreasing order by their posterior mean SUCRA value shown in the main diagonal. The larger the treatment effect, the darker the color shade. The league_heatmap function can also display the treatment effects from two outcomes.

Prediction

The league_heatmap_pred function has the same arguments as league_heatmap, but it displays predictions for all possible pairwise comparisons in the network:

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# The heatmap league table for prediction 
league_heatmap_pred(full1 = res, 
                    drug_names1 = interv_names)

Forest plot of estimation and prediction

A compact alternative to the league table is to create a forest plot of comparisons with a selected comparator intervention. For illustration, we have selected budesidone as the reference and we ran the following code to obtain the forest plot where results on estimation and prediction co-appear:

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# The heatmap league table for prediction 
forestplot(full = res, 
           compar = "BUDE",
           drug_names = abbr_interv_names)

In plot A), the 95% credible intervals (black lines) overlay the 95% prediction intervals (orange lines). In both plots, the interventions are sorted from best to worst based on their SUCRA value (see plot B)).

Intervention ranking and relevant measures

The rankosucra_plot function creates a plot for each intervention that combines two popular hierarchy measures: the rankogram and SUCRA plot:

# The names of the interventions are in the order they appear in the dataset
interv_names <- c("placebo", "budesonide", "budesonide plus formoterol", "fluticasone", "fluticasone plus salmeterol", "formoterol", "salmeterol", "tiotropium")

# The rankogram-SUCRA-plot 'amalgamation'
rankosucra_plot(full1 = res, 
                drug_names1 = interv_names)

The interventions are sorted from best to worst based on their SUCRA value (the number on the top left corner of each plot). The rankosucra_plot function can also display the hierarchy results from two outcomes; type ?rankosucra_plot for more details.

Pairwise versus network meta-analysis

The functions run_series_meta and series_meta_plot are designed to explore the implications of assuming consistency and common heterogeneity, as opposed to performing pairwise meta-analysis separately for each observed comparison in the network.

First, use the run_series_meta function to conduct separate pairwise meta-analyses for all observed comparisons in the network:

# Run separate pairwise meta-analyses
meta <- run_series_meta(full = res)

Here, the function inherits the arguments n_chains, n_iter, n_burning, and n_thin from the run_model function via res; however, you can also specify these arguments anew. Just as with the run_model, the console prints the progress of simulations together with the number of models to run: there are 15 observed comparisons in the network; hence, the console prints this message (in red), followed by a progress bar for each analysis alongside the model update until convergence:

Next, we insert meta into the function series_meta_plot to experience the visualisation toolkit for this analysis:

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# End-user-ready results for a series of pairwise meta-analyses
series_meta_plot(full = res,
                 meta = meta,
                 drug_names = abbr_interv_names)

This function returns the same results in two formats: (i) a table that is printed on the console, and (ii) a panel of forest plots.

Network meta-analysis produced more precise results than pairwise meta-analysis for all observed comparisons (plot A)). The benefits of conducting network meta-analysis for this example are also evident for the heterogeneity parameter, which was estimated at a reasonable level and with greater precision than in any pairwise meta-analysis (plot B)): blue vertical lines refer to the posterior median and 95% credible interval of the common heterogeneity parameter in network meta-analysis.

You can export the table to an xlsx file in your working directory by adding the argument save_xls = TRUE in the series_meta_plot function.

Note that comparing network meta-analysis with pairwise meta-analysis results should never be used to draw conclusions about the possibility of consistency assumption! There are specific tools designed to evaluate the consistency assumption. In the next part of the tutorial, we will introduce you to functions that offer local and global assessment of the consistency assumption.

Consistency evaluation

The node-splitting approach (local evaluation)

Do you see at least one closed-loop of interventions in your network? Are these loops informed by two-arm trials only, or a combination of two-arm and multi-arm trials? If so, you must evaluate whether network meta-analysis produces sensible results under the consistency assumption or if there are loops with evidence of possible inconsistency. Our COPD network has several closed loops (see the network plot above).

The run_nodesplit function applies the node-splitting approach with the synergy of the R package gemtc to select the proper nodes to split (read van Valkenhoef and colleagues for more details on which nodes to split when multi-arm trials are also present in the loops):

# Run the node-splitting approach
node <- run_nodesplit(full = res)

Just as with the run_series_meta function, run_nodesplit can inherit the MCMC-related arguments from the run_model function via res, or we can specify these arguments anew. Likewise, the console prints the progress of simulations and the number of node-splitting models to run. There are 8 such models (the message in red), and each one of these models will also be updated until convergence:

Next, we insert node into the function nodesplit_plot to get the necessary results both in tabular and graphical format:

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# End-user-ready results for the consistency evaluation locally
nodesplit_plot(full = res,
               node = node,
               drug_names = abbr_interv_names)

Just as with the series_meta_plot function, you can export each table to an xlsx file at your working directory by inserting the argument save_xls = TRUE in the nodesplit_plot function.

The first graph is a panel of interval plots to check consistency through the inconsistency factor (the difference between direct and indirect estimate):

For each split node, we can see the estimated direct effect (from pairwise meta-analysis), the indirect effect (after removing the corresponding comparison from the network), and the inconsistency factor (IF). The deviance information criterion for each model appear on the top left of each plot. The plots have been sorted in ascending order of the deviance information criterion. The 95% credible interval for the inconsistency factor crosses the vertical line (of consistency) in all plots. One may conclude that the consistency assumption holds; however, the point estimate is not even close to zero in all plots. Thinking hat on the head.

The next graph is a ‘reverse’ forest plot on the between-trial standard deviation after each split node:

Again, the split nodes have been sorted in ascending order of the deviance information criterion. The blue horizontal lines refer to the posterior median and 95% credible interval of the common heterogeneity parameter in network meta-analysis.

The unrelated mean effects model (global evaluation)

This approach boils down to comparing the network meta-analysis model with a series of pairwise meta-analyses that share a common heterogeneity parameter and are performed in one step. These two models are compared regarding a set of goodness of fit measuress and relevant diagnostic plots. Let’s dive in two functions devised for the global evaluation of consistency: run_ume and ume_plot.

The run_ume function conducts the unrelated mean effects model of Dias and colleagues:

# Run the unrelated means effect model
ume <- run_ume(full = res)

Next, we insert ume into the function ume_plot to shed light on the validity of consistency globally through a variety of results in tabular and graphical format:

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# End-user-ready results for the consistency evaluation globally
ume_plot(full = res,
         ume = ume,
         drug_names = abbr_interv_names)

Looking at the last two tables, we notice no material difference between the network meta-analysis and the unrelated mean effects model, and hence, the message in red at the start of the console that There is little to choose between the two models. Therefore, consistency may hold globally.

Again, you can export each table to an xlsx file at your working directory by inserting the argument save_xls = TRUE in the ume_plot function.

The first panel of graphs consists of the scatterplot (left) and the Bland-Altman plot (right) on the posterior mean deviance under the network meta-analysis and unrelated mean effects model:

The next panel includes the leverage plot under the network meta-analysis (left) and unrelated mean effects model (right):

Lastly, a panel of interval plots on the odds ratio under both models only for the pairwise comparisons that are observed in the network:

Network meta-regression

Last in the methods that evaluate the assumptions of network meta-analysis is network meta-regression. We will investigate whether the publication year may have a modification effect. In other words, do intervention effects change over time? We created the vector cov with the publication year of the trials in the order they appear in the dataset nma.baker2009.

Run the following code to conduct network meta-regression using the function run_metareg:

# The year of publication
cov <- c(1996, 1998, 1999, rep(2000, 2), 2001, rep(2002, 5), rep(2003, 2), rep(2005, 4), rep(c(2006, 2007), each = 2))

# Run the unrelated means effect model
reg <- run_metareg(full = res,
                   covariate = cov,
                   covar_assumption = "exchangeable")

As you may have guessed by now, we insert reg into the function metareg_plot to obtain several results in tabular and graphical format:

# Abbreviated interventions (in the order they appear in the datasets)
abbr_interv_names <- c("placebo", "BUDE", "BUDE plus FORM", "FLUT", "FLUT plus SALM", "FORM", "SALM", "TIOT")

# End-user-ready results for network meta-regression
metareg_plot(full = res,
             reg = reg,
             compar = "BUDE", 
             cov_value = list(2002, "Publication year"),
             drug_names = abbr_interv_names)

In case you were wondering why a simulation progress appears on the R console, the posterior mean and standard deviation of the basic parameters and corresponding regression coefficients (as estimated from run_metareg) are inserted into the BUGS code to obtain the posterior distribution of SUCRA at the selected value of the covariate (cov_value).

Overall, the network meta-regression results do not seem to differ substantially from the network meta-analysis results. The posterior mean of the regression coefficient for each intervention (last table) is another hint that the intervention effects may not change over time systematically: the ratio of odds ratios between two subsequent years almost equals 1.

As a reminder, you may insert the argument save_xls = TRUE in the metareg_plot function to export each table to an xlsx file at your working directory.

Just like the forestplot function, the 95% credible intervals (black lines) overlap the 95% prediction intervals (coloured lines) in plot A). The odds ratios refer to comparisons with budesidone, the selected comparator intervention (compar = "BUDE"). In both plots, the interventions are sorted from best to worst based on their SUCRA value (see plot B)).

The hierarchy of the interventions does not materially change when comparing the two models regarding the SUCRA values for the publication year 2002.

Conclusion

The main functionalities of rnmamod have been presented in this tutorial. The interested reader is prompted to browse through the manual to dive in the technical details of the functions.

For any questions about the tutorial and the rnmamod R package, feel free to contact Spineli.Loukia@mh-hannover.de. For any information about the author, visit loukiaspineli.netlify.app.